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This paper introduces a new subclass of generalized bi-Bazilevic-type functions associated with the generating function of generalized Mersenne polynomials, which encompasses several eminent and widely studied subclasses, such as bi-starlike and bi-convex functions. The use of generalized Mersenne polynomials highlights the applicability of special polynomials in geometric function theory. By applying the generating function of the generalized Mersenne polynomials, we derive coefficient bounds for the initial Taylor–Maclaurin coefficients and provide estimates for the Fekete–Szego inequality. Furthermore, we discuss several corollaries by specifying particular choices of the parameters.

Let ϕ(z)=zd+c be a polynomial over a field K. We study the inverse stability of ϕ(z) over K. In this paper, we establish some sufficient conditions for the inverse stability of ϕ(z) over the field of rational numbers and a function field. Furthermore, we also provide necessary and sufficient conditions for the inverse stability of ϕ(z)over a finite field.

We prove a new irreducibility result for polynomials over ${\\mathbb Q}$ and we use it to construct new infinite families of reciprocal monogenic quintinomials in ${\\mathbb Z}[x]$ of degree $2^n$ .

The connected domination polynomial of a connected graph G of order n is the polynomial D c [ G , x ] = ∑ i = γ c n d c ( G , i ) x i , where d c ( G, i ) is the number of connected dominating sets of G of cardinality i and γ c (G) is the connected domination number of G [5]. In this paper we find the polynomial D c ( G, x ) for some constructive graphs.

Recently, operational matrices were adapted for solving several kinds of fractional differential equations (FDEs). The use of numerical techniques in conjunction with operational matrices of some orthogonal polynomials, for the solution of FDEs on finite and infinite intervals, produced highly accurate solutions for such equations. This article discusses spectral techniques based on operational matrices of fractional derivatives and integrals for solving several kinds of linear and nonlinear FDEs. More precisely, we present the operational matrices of fractional derivatives and integrals, for several polynomials on bounded domains, such as the Legendre, Chebyshev, Jacobi and Bernstein polynomials, and we use them with different spectral techniques for solving the aforementioned equations on bounded domains. The operational matrices of fractional derivatives and integrals are also presented for orthogonal Laguerre and modified generalized Laguerre polynomials, and their use with numerical techniques for solving FDEs on a semi-infinite interval is discussed. Several examples are presented to illustrate the numerical and theoretical properties of various spectral techniques for solving FDEs on finite and semi-infinite intervals.

In this paper, we define p R q polynomial, which is denoted by R n ( z ) . We also discuss some generating functions and recurrence relations for R n ( z ) polynomial and its applications.

By using Nevanlinna of the value distribution of meromorphic functions, we investigate the transcendental meromorphic solutions of the non-linear differential equationfn+Pd(f)=p1eα1z+p2eα2z+p3eα3z,where Pd(f) is a differential polynomial in f of degree d(0≤d≤n−3) with small meromorphic coefficients and pi,αi(i=1,2,3) are nonzero constants. We show that the solutions of this type equation are exponential sums and they are in Γ0∪Γ1∪Γ3 which will be given in Section 1. Moreover, we give some examples to illustrate our results.

We prove a generalization of the \\(q\\)-Selberg integral evaluation formula. The integrand is that of \\(q\\)-Selberg integral multiplied by a factor of the same form with respect to part of the variables. The proof relies on the quadratic norm formula of Koornwinder polynomials. We also derive generalizations of Mehta's integral formula as limit cases of our integral.

For the purpose of resolving dual series problems involving Laguerre polynomials, this essay makes use of the Noble and Lowndes multiplying factor approach.

Random matrix theory, both as an application and as a theory, has evolved rapidly over the past fifteen years.Log-Gases and Random Matricesgives a comprehensive account of these developments, emphasizing log-gases as a physical picture and heuristic, as well as covering topics such as beta ensembles and Jack polynomials. Peter Forrester presents an encyclopedic development of log-gases and random matrices viewed as examples of integrable or exactly solvable systems. Forrester develops not only the application and theory of Gaussian and circular ensembles of classical random matrix theory, but also of the Laguerre and Jacobi ensembles, and their beta extensions. Prominence is given to the computation of a multitude of Jacobians; determinantal point processes and orthogonal polynomials of one variable; the Selberg integral, Jack polynomials, and generalized hypergeometric functions; Painlevé transcendents; macroscopic electrostatistics and asymptotic formulas; nonintersecting paths and models in statistical mechanics; and applications of random matrix theory. This is the first textbook development of both nonsymmetric and symmetric Jack polynomial theory, as well as the connection between Selberg integral theory and beta ensembles. The author provides hundreds of guided exercises and linked topics, makingLog-Gases and Random Matricesan indispensable reference work, as well as a learning resource for all students and researchers in the field.